Category Archives: Decision Theory

Kenneth Arrow Part II: The Theory of General Equilibrium

The first post in this series discussed Ken Arrow’s work in the broad sense, with particular focus on social choice. In this post, we will dive into his most famous accomplishment, the theory of general equilibrium (1954, Econometrica). I beg the reader to offer some sympathy for the approximations and simplifications that will appear below: the history of general equilibrium is, by this point, well-trodden ground for historians of thought, and the interpretation of history and theory in this area is quite contentious.

My read of the literature on GE following Arrow is as follows. First, the theory of general equilibrium is an incredible proof that markets can, in theory and in certain cases, work as efficiently as an all-powerful planner. That said, the three other hopes of general equilibrium theory since the days of Walras are, in fact, disproven by the work of Arrow and its followers. Market forces will not necessarily lead us toward these socially optimal equilibrium prices. Walrasian demand does not have empirical content derived from basic ordinal utility maximization. We cannot rigorously perform comparative statics on general equilibrium economic statistics without assumptions that go beyond simple utility maximization. From my read of Walras and the early general equilibrium theorists, all three of those results would be a real shock.

Let’s start at the beginning. There is an idea going back to Adam Smith and the invisible hand, an idea that individual action will, via the price system, lead to an increase or even maximization of economic welfare (an an aside, Smith’s own use of “invisible hand” trope is overstated, as William Grampp among others has convincingly argued). The kind of people who denigrate modern economics – the neo-Marxists, the back-of-the-room scribblers, the wannabe-contrarian-dilletantes – see Arrow’s work, and the idea of using general equilibrium theory to “prove that markets work”, as a barbarism. We know, and have known well before Arrow, that externalities exist. We know, and have known well before Arrow, that the distribution of income depends on the distribution of endowments. What Arrow was interested in was examining not only whether the invisible hand argument “is true, but whether it could be true”. That is, if we are to claim markets are uniquely powerful at organizing economic activity, we ought formally show that the market could work in such a manner, and understand the precise conditions under which it won’t generate these claimed benefits. How ought we do this? Prove the precise conditions under which there exists a price vector where markets clear, show the outcome satisfies some welfare criterion that is desirable, and note exactly why each of the conditions are necessary for such an outcome.

The question is, how difficult is it to prove these prices exist? The term “general equilibrium” has had many meanings in economics. Today, it is often used to mean “as opposed to partial equilibrium”, meaning that we consider economic effects allowing all agents to adjust to a change in the environment. For instance, a small random trial of guaranteed incomes has, as its primary effect, an impact on the incomes of the recipients; the general equilibrium effects of making such a policy widespread on the labor market will be difficult to discern. In the 19th and early 20th century, however, the term was much more concerned with the idea of the economy as a self-regulating system. Arrow put it very nicely in an encyclopedia chapter he wrote in 1966: general equilibrium is both “the simple notion of determinateness, that the relations which describe the economic system must form a system sufficiently complete to determine the values of its variables and…the more specific notion that each relation represents a balance of forces.”

If you were a classical, a Smith or a Marx or a Ricardo, the problem of what price will obtain in a market is simple to solve: ignore demand. Prices are implied by costs and a zero profit condition, essentially free entry. And we more or less think like this now in some markets. With free entry and every firm producing at the identical minimum efficient scale, price is entirely determined by the supply side, and only quantity is determined by demand. With one factor, labor where the Malthusian condition plays the role of free entry, or labor and land in the Ricardian system, this classical model of value is well-defined. How to handle capital and differentiated labor is a problem to be assumed away, or handled informally; Samuelson has many papers where he is incensed by Marx’s handling of capital as embodied labor.

The French mathematical economist Leon Walras finally cracked the nut by introducing demand and price-taking. There are household who produce and consume. Equilibrium involves supply and demand equating in each market, hence price is where margins along the supply and demand curves equate. Walras famously (and informally) proposed a method by which prices might actually reach equilibrium: the tatonnement. An auctioneer calls out a price vector: in some markets there is excess demand and in some excess supply. Prices are then adjusted one at a time. Of course each price change will affect excess demand and supply in other markets, but you might imagine things can “converge” if you adjust prices just right. Not bad for the 1870s – there is a reason Schumpeter calls this the “Magna Carta” of economic theory in his History of Economic Analysis. But Walras was mistaken on two counts: first, knowing whether there even exists an equilibrium that clears every market simultaneously is, it turns out, equivalent to a problem in Poincare’s analysis situs beyond the reach of mathematics in the 19th century, and second, the conditions under which tatonnement actually converges are a devilish problem.

The equilibrium existence problem is easy to understand. Take the simplest case, with all j goods made up of the linear combination of k factors. Demand equals supply just says that Aq=e, where q is the quantity of each good produced, e is the endowment of each factor, and A is the input-output matrix whereby product j is made up of some combination of factors k. Also, zero profit in every market will imply Ap(k)=p(j), where p(k) are the factor prices and p(j) the good prices. It was pointed out that even in this simple system where everything is linear, it is not at all trivial to ensure that prices and quantities are not negative. It would not be until Abraham Wald in the mid-1930s – later Arrow’s professor at Columbia and a fellow Romanian, links that are surely not a coincidence! – that formal conditions were shown giving existence of general equilibrium in a simple system like this one, though Wald’s proof greatly simplified by the general problem by imposing implausible restrictions on aggregate demand.

Mathematicians like Wald, trained in the Vienna tradition, were aghast at the state of mathematical reasoning in economics at the time. Oskar Morgenstern absolutely hammered the great economist John Hicks in a 1941 review of Hicks’ Value and Capital, particularly over the crazy assertion (similar to Walras!) that the number of unknowns and equations being identical in a general equilibrium system sufficed for a solution to exist (if this isn’t clear to you in a nonlinear system, a trivial example with two equations and two unknowns is here). Von Neumann apparently said (p. 85) to Oskar, in reference to Hicks and those of his school, “if those books are unearthed a hundred years hence, people will not believe they were written in our time. Rather they will think they are about contemporary with Newton, so primitive is the mathematics.” And Hicks was quite technically advanced compared to his contemporary economists, bringing the Keynesian macroeconomics and the microeconomics of indifference curves and demand analysis together masterfully. Arrow and Hahn even credit their initial interest in the problems of general equilibrium to the serendipity of coming across Hicks’ book.

Mathematics had advanced since Walras, however, and those trained at the mathematical frontier finally had the tools to tackle Walras’ problem seriously. Let D(p) be a vector of demand for all goods given price p, and e be initial endowments of each good. Then we simply need D(p)=e or D(p)-e=0 in each market. To make things a bit harder, we can introduce intermediate and factor goods with some form of production function, but the basic problem is the same: find whether there exists a vector p such that a nonlinear equation is equal to zero. This is the mathematics of fixed points, and Brouwer had, in 1912, given a nice theorem: every continuous function from a compact convex subset to itself has a fixed point. Von Neumann used this in the 1930s to prove a similar result to Wald. A mathematician named Shizuo Kakutani, inspired by von Neumann, extended the Brouwer result to set-valued mappings called correspondences, and John Nash in 1950 used that result to show, in a trivial proof, the existence of mixed equilibria in noncooperative games. The math had arrived: we had the tools to formally state when non-trivial non-linear demand and supply systems had a fixed point, and hence a price that cleared all markets. We further had techniques for handling “corner solutions” where demand for a given good was zero at some price, surely a common outcome in the world: the idea of the linear program and complementary slackness, and its origin in convex set theory as applied to the dual, provided just the mathematics Arrow and his contemporaries would need.

So here we stood in the early 1950s. The mathematical conditions necessary to prove that a set-valued function has an equilibrium have been worked out. Hicks, in Value and Capital, has given Arrow the idea that relating the future to today is simple: just put a date on every commodity and enlarge the commodity space. Indeed, adding state-contingency is easy: put an index for state in addition to date on every commodity. So we need not only zero excess demand in apples, or in apples delivered in May 1955, but in apples delivered in May 1955 if Eisenhower loses his reelection bid. Complex, it seems, but no matter: the conditions for the existence of a fixed point will be the same in this enlarged commodity space.

With these tools in mind, Arrow and Debreu can begin their proof. They first define a generalization of an n-person game where the feasible set of actions for each player depends on the actions of every other player; think of the feasible set as “what can I afford given the prices that will result for the commodities I am endowed with?” The set of actions is an n-tuple where n is the number of date and state indexed commodities a player could buy. Debreu showed in 1952 PNAS that these generalized games have an equilibrium as long as each payoff function varies continuously with other player’s actions, the feasible set of choices convex and varies continuously in other player’s actions, and the set of actions which improve a player’s payoff are convex for every action profile. Arrow and Debreu then show that the usual implications on individual demand are sufficient to aggregate up to the conditions Debreu’s earlier paper requires. This method is much, much different from what is done by McKenzie or other early general equilibrium theorists: excess demand is never taken as a primitive. This allows the Arrow-Debreu proof to provide substantial economic intuition as Duffie and Sonnenschein point out in a 1989 JEL. For instance, showing that the Arrow-Debreu equilibrium exists even with taxation is trivial using their method but much less so in methods that begin with excess demand functions.

This is already quite an accomplishment: Arrow and Debreu have shown that there exists a price vector that clears all markets simultaneously. The nature of their proof, as later theorists will point out, relies less on convexity on preferences and production sets as on the fact that every agent is “small” relative to the market (convexity is used to get continuity in the Debreu game, and you can get this equally well by making all consumers infinitesimal and then randomizing allocations to smooth things out; see Duffie and Sonnenschein above for an example). At this point, it’s the mid-1950s, heyday of the Neoclassical synthesis: surely we want to be able to answer questions like, when there is a negative demand shock, how will the economy best reach a Pareto-optimal equilibrium again? How do different speeds of adjustment due to sticky prices or other frictions affect the rate at which optimal is regained? Those types of question implicitly assume that the equilibrium is unique (at least locally) so that we actually can “return” to where we were before the shock. And of course we know some of the assumptions needed for the Arrow-Debreu proof are unrealistic – e.g., no fixed costs in production – but we would at least like to work out how to manipulate the economy in the “simple” case before figuring out how to deal with those issues.

Here is where things didn’t work out as hoped. Uzawa (RESTUD, 1960) proved that not only could Brouwer’s theorem be used to prove the existence of general equilibrum, but that the opposite was true as well: the existence of general equilibrium was logically equivalent to Brouwer. A result like this certainly makes one worry about how much one could say about prices in general equilibrium. The 1970s brought us the Sonnenschein-Mantel-Debreu “Anything Goes” theorem: aggregate excess demand functions do not inherit all the properties of individual excess demand functions because of wealth effects (when relative prices change, the value of one’s endowment changes as well). For any aggregate excess demand function satisfying a couple minor restrictions, there exists an economy with individual preferences generating that function; in particular, fewer restrictions than are placed on individual excess demand as derived from individual preference maximization. This tells us, importantly, that there is no generic reason for equilibria to be unique in an economy.

Multiplicity of equilibria is a problem: if the goal of GE was to be able to take underlying primitives like tastes and technology, calculate “the” prices that clear the market, then examine how those prices change (“comparative statics”), we essentially lose the ability to do all but local comparative statics since large changes in the environment may cause the economy to jump to a different equilibrium (luckily, Debreu (1970, Econometrica) at least generically gives us a finite number of equilibria, so we may at least be able to say something about local comparative statics for very small shocks). Indeed, these analyses are tough without an equilibrium selection mechanism, which we don’t really have even now. Some would say this is no big deal: of course the same technology and tastes can generate many equilibria, just as cars may wind up all driving on either the left or the right in equilibrium. And true, all of the Arrow-Debreu equilibria are Pareto optimal. But it is still far afield from what might have been hoped for in the 1930s when this quest for a modern GE theory began.

Worse yet is stability, as Arrow and his collaborators (1958, Ecta; 1959, Ecta) would help discover. Even if we have a unique equilibrium, Herbert Scarf (IER, 1960) showed, via many simple examples, how Walrasian tatonnement can lead to cycles which never converge. Despite a great deal of the intellectual effort in the 1960s and 1970s, we do not have a good model of price adjustment even now. I should think we are unlikely to ever have such a theory: as many theorists have pointed out, if we are in a period of price adjustment and not in an equilibrium, then the zero profit condition ought not apply, ergo why should there be “one” price rather than ten or a hundred or a thousand?

The problem of multiplicity and instability for comparative static analysis ought be clear, but it should also be noted how problematic they are for welfare analysis. Consider the Second Welfare Theorem: under the Arrow-Debreu system, for every Pareto optimal allocation, there exists an initial endowment of resources such that that allocation is an equilibrium. This is literally the main justification for the benefits of the market: if we reallocate endowments, free exchange can get us to any Pareto optimal point, ergo can get us to any reasonable socially optimal point no matter what social welfare function you happen to hold. How valid is this justification? Call x* the allocation that maximizes some social welfare function. Let e* be an initial endowment for which x* is an equilibrium outcome – such an endowment must exist via Arrow-Debreu’s proof. Does endowing agents with e* guarantee we reach that social welfare maximum? No: x* may not be unique. Even if it unique, will we reach it? No: if it is not a stable equilibrium, it is only by dint of luck that our price adjustment process will ever reach it.

So let’s sum up. In the 1870s, Walras showed us that demand and supply, with agents as price takers, can generate supremely useful insights into the economy. Since demand matters, changes in demand in one market will affect other markets as well. If the price of apples rises, demand for pears will rise, as will their price, whose secondary effect should be accounted for in the market for apples. By the 1930s we have the beginnings of a nice model of individual choice based on constrained preference maximization. Taking prices as given, individual demands have well-defined forms, and excess demand in the economy can be computed by a simple summing up. So we now want to know: is there in fact a price that clears the market? Yes, Arrow and Debreu show, there is, and we needn’t assume anything strange about individual demand to generate this. These equilibrium prices always give Pareto optimal allocations, as had long been known, but there also always exist endowments such that every Pareto optimal allocation is an equilibria. It is a beautiful and important result, and a triumph for the intuition of the invisible hand it its most formal sense.

Alas, it is there we reach a dead end. Individual preferences alone do not suffice to tell us what equilibria we are at, nor that any equilibria will be stable, nor that any equilibria will be reached by an economically sensible adjustment process. To say anything meaningful about aggregate economic outcomes, or about comparative statics after modest shocks, or about how technological changes change price, we need to make assumptions that go beyond individual rationality and profit maximization. This is, it seems to me, a shock for the economists of the middle of the century, and still a shock for many today. I do not think this means “general equilibrium is dead” or that the mathematical exploration in the field was a waste. We learned a great deal about precisely when markets could even in principle achieve the first best, and that education was critical for the work Arrow would later do on health care, innovation, and the environment, which I will discuss in the next two posts. And we needn’t throw out general equilibrium analysis because of uniqueness or stability problems, any more than we would throw out game theoretic analysis because of the same problems. But it does mean that individual rationality as the sole paradigm of economic analysis is dead: it is mathematically proven that postulates of individual rationality will not allow us to say anything of consequence about economic aggregates or game theoretic outcomes in the frequent scenarios where we do not have a unique equilibria with a well-defined way to get there (via learning in games, or a tatonnament process in GE, or something of a similar nature). Arrow himself (1986, J. Business) accepts this: “In the aggregate, the hypothesis of rational behavior has in general no implications.” This is an opportunity for economists, not a burden, and we still await the next Arrow who can guide us on how to proceed.

Some notes on the literature: For those interested in the theoretical development of general equilibrium, I recommend General Equilibrium Analysis by Roy Weintraub, a reformed theorist who now works in the history of thought. Wade Hands has a nice review of the neoclassical synthesis and the ways in which Keynesianism and GE analysis were interrelated. On the battle for McKenzie to be credited alongside Arrow and Debreu, and the potentially scandalous way Debreu may have secretly been responsible for the Arrow and Debreu paper being published first, see the fine book Finding Equilibrium by Weintraub and Duppe; both Debreu and McKenzie have particularly wild histories. Till Duppe, a scholar of Debreu, also has a nice paper in the JHET on precisely how Arrow and Debreu came to work together, and what the contribution of each to their famous ’54 paper was.

The Greatest Living Economist Has Passed Away: Notes on Kenneth Arrow Part I

It is amazing how quickly the titans of the middle of the century have passed. Paul Samuelson and his mathematization, Ronald Coase and his connection of law to economics, Gary Becker and his incorporation of choice into the full sphere of human behavior, John Nash and his formalization of strategic interaction, Milton Friedman and his defense of the market in the precarious post-war period, Robert Fogel and his cliometric revolution: the remaining titan was Kenneth Arrow, the only living economist who could have won a second Nobel Prize without a whit of complaint from the gallery. These figures ruled as economics grew from a minor branch of moral philosophy into the most influential, most prominent, and most advanced of the social sciences. It is hard to imagine our field will ever again have such a collection of scholars rise in one generation, and with the tragic news that Ken has now passed away as well, we have, with great sadness and great rapidity, lost the full set.

Though he was 95 years old, Arrow was still hard at work; his paper with Kamran Bilir and Alan Sorensen was making its way around the conference circuit just last year. And beyond incredible productivity, Arrow had a legendary openness with young scholars. A few years ago, a colleague and I were debating a minor point in the history of economic thought, one that Arrow had played some role in; with the debate deadlocked, it was suggested that I simply email the protagonist to learn the truth. No reply came; perhaps no surprise, given how busy he was and how unknown I was. Imagine my surprise when, two months letter, a large manila envelope showed up in my mailbox at Northwestern, with a four page letter Ken had written inside! Going beyond a simple answer, he patiently walked me through his perspective on the entire history of mathematical economics, the relative centrality of folks like Wicksteed and Edgeworth to the broader economic community, the work he did under Hotelling and the Cowles Commission, and the nature of formal logic versus price theory. Mind you, this was his response to a complete stranger.

This kindness extended beyond budding economists: Arrow was a notorious generator of petitions on all kinds of social causes, and remained so late in life, signing the Economists Against Trump that many of us supported last year. You will be hardpressed to find an open letter or amicus curiae, on any issue from copyright term extension to the use of nuclear weapons, which Arrow was unaware of. The Duke Library holds the papers of both Arrow and Paul Samuelson – famously they became brothers-in-law – and the frequency with which their correspondence involves this petition or that, with Arrow in general the instigator and Samuelson the deflector, is unmistakable. I recall a great series of letters where Arrow queried Samuelson as to who had most deserved the Nobel but had died too early to receive it. Arrow at one point proposed Joan Robinson, which sent Samuelson into convulsions. “But she was a communist! And besides, her theory of imperfect competition was subpar.” You get the feeling in these letters of Arrow making gentle comments and rejoinders while Samuelson exercises his fists in the way he often did when battling everyone from Friedman to the Marxists at Cambridge to (worst of all, for Samuelson) those who were ignorant of their history of economic thought. Their conversation goes way back: you can find in one of the Samuelson boxes his recommendation that the University of Michigan bring in this bright young fellow named Arrow, a missed chance the poor Wolverines must still regret!

Arrow is so influential, in some many areas of economics, that it is simply impossible to discuss his contributions in a single post. For this reason, I will break the post into four parts, with one posted each day this week. We’ll look at Arrow’s work in choice theory today, his work on general equilibrium tomorrow, his work on innovation on Thursday, and some selected topics where he made seminal contributions (the economics of the environment, the principal-agent problem, and the economics of health care, in particular) on Friday. I do not lightly say that Arrow was the greatest living economist, and in my reckoning second only to Samuelson for the title of greatest economist of all time. Arrow wrote the foundational paper of general equilibrium analysis, the foundational paper of social choice and voting, the foundational paper justifying government intervention in innovation, and the foundational paper in the economics of health care. His legacy is the greatest legacy possible for the mathematical approach pushed by the Cowles Commission, the Econometric Society, Irving Fisher, and the mathematician-cum-economist Harold Hotelling. And so it is there that we must begin.

Arrow was born in New York City, a CCNY graduate like many children of the Great Depression, who went on to study mathematics in graduate school at Columbia. Economics in the United States in the 1930s was not a particularly mathematical science. The formalism of von Neumann, the late-life theoretical conversion of Schumpeter, Samuelson’s Foundations, and the soft nests at Cowles and the Econometric Society were in their infancy.

The usual story is that Arrow’s work on social choice came out of his visit to RAND in 1948. But this misstates the intellectual history: Arrow’s actual encouragement comes from his engagement with a new form of mathematics, the expansions of formal logic beginning with people like Peirce and Boole. While a high school student, Arrow read Bertrand Russell’s text on mathematical logic, and was enthused with the way that set theory permitted logic to go well beyond the syllogisms of the Greeks. What a powerful tool for the generation of knowledge! His Senior year at CCNY, Arrow took the advanced course on relational logic taught by Alfred Tarski, where the eminent philosopher took pains to reintroduce the ideas of Charles Sanders Peirce, the greatest yet most neglected American philosopher. The idea of relations are familiar to economists: give some links between a set (i.e, xRy and yRz) and some properties to the relation (i.e., it is well-ordered), and you can then perform logical operations on the relation to derive further properties. Every trained economist sees an example of this when first learning about choice and utility, but of course things like “greater than” and “less than” are relations as well. In 1940, one would have had to be extraordinarily lucky to encounter this theory: Tarski’s own books were not even translated.

But what great training this would be! For Arrow joined a graudate program in mathematical statistics at Columbia, where one of the courses was taught by Hotelling from the economics department. Hotelling was an ordinalist, rare in those days, and taught his students demand theory from a rigorous basis in ordinal preferences. But what are these? Simply relations with certain properties! Combined with a statistician’s innate ability to write proofs using inequalities, Arrow greatly impressed Hotelling, and switched to a PhD in economics with inspiration in the then-new subfield on mathematical economics that Hotelling, Samuelson, and Hicks were helping to expand.

After his wartime service doing operation research related to weather and flight planning, and a two year detour into capital theory with little to show for it, Arrow took a visiting position at the Cowles Commission, a center of research in mathematical economics then at the University of Chicago. In 1948, Arrow spent the summer at RAND, still yet to complete his dissertation, or even to strike on a worthwhile idea. RAND in Santa Monica was the world center for applied game theory: philosophers, economists, and mathematicians prowled the halls working through the technical basics of zero-sum games, but also the application of strategic decision theory to problems of serious global importance. Arrow had been thinking about voting a bit, and had written a draft of a paper, similar to that of Duncan Black’s 1948 JPE, essentially suggesting that majority voting “works” when preferences are single-peaked; that is, if everyone can rank options from “left to right”, and simply differ on which point is their “peak” of preference, then majority voting reflects individual preferences in a formal sense. At RAND, the philosopher Olaf Helmer pointed out that a similar concern mattered in international relations: how are we to say that the Soviet Union or the United States have preferences? They are collections of individuals, not individuals themselves.

Right, Arrow agreed. But economists had thought about collective welfare, from Pareto to Bergson-Samuelson. The Bergson-Samuelson idea is simple. Let all individuals in society have preferences over states of the world. If we all prefer state A to state B, then the Pareto criterion suggests society should as well. Of course, tradeoffs are inevitable, so what are we to do? We could assume cardinal utility (e.g., “how much money are willing to be paid to accept A if you prefer B to A and society goes toward A?”) as in the Kaldor-Hicks criterion (though the technically minded will know that Kaldor-Hicks does not define an order on states of the world, so isn’t really great for social choice). But let’s assume all people have is their own ordinal utility, their own rank-order of states, an order that is naturally hard to compare across people. Let’s assume for some pairs we have Pareto dominance: we all prefer A to C, and Q to L, and Z to X, but for other pairs there is no such dominance. A great theorem due to the Polish mathematician Szpilrain, and I believe popularized among economists by Blackwell, says that if you have a quasiorder R that is transitive, then there exists an order R’ which completes it. In simple terms, if you can rank some pairs, and the pairs you do rank do not have any intransitivity, then you can generate a complete rankings of all pairs which respects the original incomplete ordering. Since individuals have transitive preferences, Pareto ranks are transitive, and hence we know there exist social welfare functions which “extend” Pareto. The implications of this are subtle: for instance, as I discuss in the link earlier in this paragraph, it implies that pure monetary egalitarianism can never be socially optimal even if the only requirement is to respect Pareto dominance.

So aren’t we done? We know what it means, via Bergson-Samuelson, for the Soviet Union to “prefer” X to Y. But alas, Arrow was clever and attacked the problem from a separate view. His view was to, rather than taking preference orderings of individuals as given and constructing a social ordering, to instead ask whether there is any mechanism for constructing a social ordering from arbitrary individual preferences that satisfies certain criteria. For instance, you may want to rule out a rule that says “whatever Kevin prefers most is what society prefers, no matter what other preferences are” (non-dictatorship). You may want to require Pareto dominance to be respected so that if everyone likes A more than B, A must be chosen (Pareto criterion). You may want to ensure that “irrelevant options” do not matter, so that if giving an option to choose “orange” in addition to “apple” and “pear” does not affect any individual’s ranking of apples and pears, then the orange option also oughtn’t affect society’s rankings of apples and pears (IIA). Arrow famously proved that if we do not restrict what types of preferences individuals may have over social outcomes, there is no system that can rank outcomes socially and still satisfy those three criteria. It has been known that majority voting suffers a problem of this sort since Condorcet in the 18th century, but the general impossibility was an incredible breakthrough, and a straightforward one once Arrow was equipped with the ideas of relational logic.

It was with this result, in the 1951 book-length version of the idea, that social choice as a field distinct from welfare economics really took off. It is a startling result in two ways. First, in pure political theory, it rather simply killed off two centuries of blather about what the “best” voting system was: majority rule, Borda counts, rank-order voting, or whatever you like, every system must violate one of the Arrow axioms. And indeed, subsequent work has shown that the axioms can be relaxed and still generate impossibility. In the end, we do need to make social choices, so what should we go with? If you’re Amartya Sen, drop the Pareto condition. Others have quibbled with IIA. The point is that there is no right answer. The second startling implication is that welfare economics may be on pretty rough footing. Kaldor-Hicks conditions, which in practice motivate all sorts of regulatory decisions in our society, both rely on the assumption of cardinal or interpersonally-comparable utility, and do not generate an order over social options. Any Bergson-Samuelson social welfare function, a really broad class, must violate some pretty natural conditions on how they treat “equivalent” people (see, e.g., Kemp and Ng 1976). One questions whether we are back in the pre-Samuelson state where, beyond Pareto dominance, we can’t say much with any rigor about whether something is “good” or “bad” for society without dictatorially imposing our ethical standard, individual preferences be damned. Arrow’s theorem is a remarkable achievement for a man as young as he was when he conceived it, one of those rare philosophical ideas that will enter the canon alongside the categorical imperative or Hume on induction, a rare idea that will without question be read and considered decades and centuries hence.

Some notes to wrap things up:

1) Most call the result “Arrow’s Impossibility Theorem”. After all, he did prove the impossibility of a certain form of social choice. But Tjalling Koopmans actually convinced Arrow to call the theorem a “Possibility Theorem” out of pure optimism. Proof that the author rarely gets to pick the eventual name!

2) The confusion between Arrow’s theorem and the existence of social welfare functions in Samuelson has a long and interesting history: see this recent paper by Herrada Igersheim. Essentially, as I’ve tried to make clear in this post, Arrow’s result does not prove that Bergson-Samuelson social welfare functions do not exist, but rather implicitly imposes conditions on the indifference curves which underlie the B-S function. Much more detail in the linked paper.

3) So what is society to do in practice given Arrow? How are we to decide? There is much to recommend in Posner and Weyl’s quadratic voting when preferences can be assumed to have some sort of interpersonally comparable cardinal structure, yet are unknown. When interpersonal comparisons are impossible and we do not know people’s preferences, the famous Gibbard-Satterthwaite Theorem says that we have no voting system that can avoid getting people to sometimes vote strategically. We might then ask, ok, fine, what voting or social choice system works “the best” (e.g., satisfies some desiderata) over the broadest possible sets of individual preferences? Partha Dasgupta and Eric Maskin recently proved that, in fact, good old fashioned majority voting works best! But the true answer as to the “best” voting system depends on the distribution of underlying preferences you expect to see – it is a far less simple question than it appears.

4) The conditions I gave above for Arrow’s Theorem are actually different from the 5 conditions in the original 1950 paper. The reason is that Arrow’s original proof is actually incorrect, as shown by Julian Blau in a 1957 Econometrica. The basic insight of the proof is of course salvageable.

5) Among the more beautiful simplifications of Arrow’s proof is Phil Reny’s “side by side” proof of Arrow and Gibbard-Satterthwaite, where he shows just how related the underlying logic of the two concepts is.

We turn to general equilibrium theory tomorrow. And if it seems excessive to need four days to cover the work on one man – even in part! – that is only because I understate the breadth of his contributions. Like Samuelson’s obscure knowledge of Finnish ministers which I recounted earlier this year, Arrow’s breadth of knowledge was also notorious. There is a story Eric Maskin has claimed to be true, where some of Arrow’s junior colleagues wanted to finally stump the seemingly all-knowing Arrow. They all studied the mating habits of whales for days, and then, when Arrow was coming down the hall, faked a vigorous discussion on the topic. Arrow stopped and turned, remaining silent at first. The colleagues had found a topic he didn’t fully know! Finally, Arrow interrupted: “But I thought Turner’s theory was discredited by Spenser, who showed that the supposed homing mechanism couldn’t possibly work”! And even this intellectual feat hardly matches Arrow’s well-known habit of sleeping through the first half of seminars, waking up to make the most salient point of the whole lecture, then falling back asleep again (as averred by, among others, my colleague Joshua Gans, a former student of Ken’s).

Some Results Related to Arrow’s Theorem

Arrow’s (Im)possibility Theorem is, and I think this is universally acknowledged, one of the great social science theorems of all time. I particularly love it because of its value when arguing with Popperians and other anti-theory types: the theorem is “untestable” in that it quite literally does not make any predictions, yet surely all would consider it a valuable scientific insight.

In this post, I want to talk about a couple. new papers using Arrow’s result is unusual ways. First, a philosopher has shown exactly how Arrow’s result is related to the general philosophical problem of choosing which scientific theory to accept. Second, a pair of computer scientists have used AI techniques to generate an interesting new method for proving Arrow.

The philosophic problem is the following. A good theory should satisfy a number of criteria; for Kuhn, these included accuracy, consistency, breadth, simplicity and fruitfulness. Imagine now there are a group of theories (about, e.g., how galaxies form, why birds have wings, etc.) and we ordinally rank them based on these criteria. Also imagine that we have ranked each theory according to these criteria and we all agree on the rankings. Which theory ought we accept? Arrow applied to theory choice gives us the worrying result that not only is there no unique method of choosing among theories but also that there may not exist any such method at all, at least if we want to satisfy unanimity, non-dictatorship and independence of irrelevant alternatives. That is, even if you and I all agree about how each theory ranks according to different desirability criteria, we still don’t have a good, general method of aggregating among criteria.

So what to do? Davide Rizza, in a new paper in Synthese (gated, I’m afraid), discusses a number of solutions. Of course, if we have more than just ordinal information about each criterion, then we can construct aggregated orders. For instance, if we assigned a number for the relative rankings on each criterion, we could just add these up for each theory and hence have an order. Note that this theory choice rule can be done even if we just have ordinal data – if there are N theories, then on criterion C, give the best theorem in that criterion N points, the second best N-1, and so on, then add up the scores. This is the famous Borda Count.

Why can’t we choose theories by the Borda Count or similar, then? Well, Borda (and any other rule that could construct an aggregate order while satisfying unanimity and non-dictatorship) must be violating the IIA assumption in Arrow. Unanimity, which insists a rule accept a theory if it considered best along every criterion, and non-dictatorship, where more than one criterion can at least matter in principle, seem totally unobjectionable. So maybe we ought just toss IIA from our theory choice rule, as perhaps Donald Saari would wish us to do. And IIA is a bit strange indeed. If I rank A>B>C, and if you require me to have transitive preferences, then just knowing the binary rankings A>B and B>C is enough to tell you that I prefer A>C even if I didn’t know that particular binary relationship. In this case, adding B isn’t “irrelevant”; there is information in the binary pairs generated by transitivity which IIA does not allow me to take advantage of. Some people call the IIA assumption “binary independence” since it aggregates using only binary relations, an odd thing given that the individual orders contain, by virtue of being orders, more than just binary relations. It turns out that there are aggregation rules which generate an order if we loosen IIA to an alternative restriction on how to use information in sequences. IIA, rather than ordinal rankings across criteria, is where Arrow poses a problem for theory choice. Now, Rizza points out that these aggregation rules needn’t be unique so we still can have situations where we all agree about how different theories rank according to each criterion, and agree on the axiomatic properties we want in an aggregation rules, yet nonetheless disagree about which theory to accept. Still worrying, though not for Kuhn, and certainly not for us crazier Feyerabend and Latour fans!

(A quick aside: How strange it is that Arrow’s Theorem is so heavily associated with voting? That every voting rule is subject to tactical behavior is Gibbard-Satterthwaite, not Arrow, and this result about strategic voting imposes nothing like an IIA assumption. Arrow’s result is about the far more general problem of aggregating orders, a problem which fundamentally has nothing to do with individual behavior. Indeed, I seem to recall that Arrow came up with his theorem while working one summer as a grad student at RAND on the problem of what, if anything, it could mean for a country to have preferences when voting on behalf of its citizens in bodies like the UN. The story also goes that when he showed his advisor – perhaps Hotelling? – what he had been working on over the summer, he was basically told the result was so good that he might as well just graduate right away!)

The second paper today comes from two computer scientists. There are lots of proofs of Arrow’s theorem – the original proof in Arrow’s 1951 book is actually incorrect! – but the CS guys use a technique I hadn’t seen before. Essentially, they first prove with a simple induction that iff you can find a case with 2 voters and 3 options that satisfies the Arrow axioms, can you find such a case with N>=2 voters and M>=3 options. This doesn’t actually narrow the problem a great deal: there are still 3!=6 ways to order 3 options, hence 6^2=36 permutations of the joint vote of the 2 voters, hence 6^36 functions mapping the voter orders to a social order. Nonetheless, the problem is small enough to be tackled by a Constraint Satisfaction algortihm which checks IIA and unanimity and finds only two social welfare functions not violating one of those constraints, which are just the cases where Agents 1 and 2 are dictators. Their algorithm took one second to run on a standard computer (clearly they are better algorithm writers than the average economist!). Sen’s theorem and Muller-Satterthwaite can also be proven using a similar restriction to the base case followed by algorithmic search.

Of course, algorithmic proofs tend to lack the insight and elegance of standard proofs. But they have benefits as well. Just as you can show that only 2 social welfare functions with N=2 voters and M=3 options satisfy IIA and unanimity, you can also show that only 94 (out of 6^36!) satisfy IIA. That is, it is IIA rather than other assumptions which is doing most of the work in Arrow. Inspecting those 94 remaining social welfare functions by hand can help elucidate alternative sets of axioms which also generate aggregation possibility or impossibility.

(And a third paper, just for fun: it turns out that Kiribati and Nauru actually use Borda counts in their elections, and that there does appear to be strategic candidate nomination behavior designed to take advantage of the non-IIA nature of Borda! IIA looks in many ways like a restriction on tactical behavior by candidates or those nominating issues, rather than a restriction on tactical behavior by voters. If you happen to teach Borda counts, this is a great case to give students.)

“Epistemic Game Theory,” E. Dekel & M. Siniscalchi (2014)

Here is a handbook chapter that is long overdue. The theory of epistemic games concerns a fairly novel justification for solution concepts under strategic uncertainty – that is, situations where what I want to do depends on other people do, and vice versa. We generally analyze these as games, and have a bunch of equilibrium (Nash, subgame perfection, etc.) and nonequilibrium (Nash bargain, rationalizability, etc.) solution concepts. So which should you use? I can think of four classes of justification for a game solution. First, the solution might be stable: if you told each player what to do, no one person (or sometimes group) would want to deviate. Maskin mentions this justification is particularly worthy when it comes to mechanism design. Second, the solution might be the outcome of a dynamic selection process, such as evolution or a particular learning rule. Third, the solution may be justified by certain axiomatic first principles; Shapley value is a good example in this class. The fourth class, however, is the one we most often teach students: a solution concept is good because it is justified by individual behavior assumptions. Nash, for example, is often thought to be justified by “rationality plus correct beliefs”. Backward induction is similarly justified by “common knowledge of rationality at all states.”

Those are informal arguments, however. The epistemic games (or sometimes, “interactive epistemology”) program seeks to formally analyze assumptions about the knowledge and rationality of players and what it implies for behavior. There remain many results we don’t know (for instance, I asked around and could only come up with one paper on the epistemics of coalitional games), but the results proven so far are actually fascinating. Let me give you three: rationality and common belief in rationality implies rationalizable strategies are played, the requirements for Nash are different depending on how players there are, and backward induction is surprisingly difficult to justify on epistemic grounds.

First, rationalizability. Take a game and remove any strictly dominated strategy for each player. Now in the reduced game, remove anything that is strictly dominated. Continue doing this until nothing is left to remove. The remaining strategies for each player are “rationalizable”. If players can hold any belief they want about what potential “types” opponents may be – where a given (Harsanyi) type specifies what an opponent will do – then as long as we are all rational, we all believe the opponents are rational, we all believe the opponents all believe that we all are rational, ad infinitum, the only possible outcomes to the game are the rationalizable ones. Proving this is actually quite complex: if we take as primitive the “hierarchy of beliefs” of each player (what do I believe my opponents will do, what do I believe they believe I will do, and so on), then we need to show that any hierarchy of beliefs can be written down in a type structure, then we need to be careful about how we define “rational” and “common belief” on a type structure, but all of this can be done. Note that many rationalizable strategies are not Nash equilibria.

So what further assumptions do we need to justify Nash? Recall the naive explanation: “rationality plus correct beliefs”. Nash takes us from rationalizability, where play is based on conjectures about opponent’s play, to an equilibrium, where play is based on correct conjectures. But which beliefs need to be correct? With two players and no uncertainty, the result is actually fairly straightforward: if our first order beliefs are (f,g), we mutually believe our first order beliefs are (f,g), and we mutually believe we are rational, then beliefs (f,g) represent a Nash equilibrium. You should notice three things here. First, we only need mutual belief (I know X, and you know I know X), not common belief, in rationality and in our first order beliefs. Second, the result is that our first-order beliefs are that a Nash equilibrium strategy will be played by all players; the result is about beliefs, not actual play. Third, with more than two players, we are clearly going to need assumptions about how my beliefs about our mutual opponent are related to your beliefs; that is, Nash will require more, epistemically, than “basic strategic reasoning”. Knowing these conditions can be quite useful. For instance, Terri Kneeland at UCL has investigated experimentally the extent to which each of the required epistemic conditions are satisfied, which helps us to understand situations in which Nash is harder to justify.

Finally, how about backward induction? Consider a centipede game. The backward induction rationale is that if we reached the final stage, the final player would defect, hence if we are in the second-to-last stage I should see that coming and defect before her, hence if we are in the third-to-last stage she will see that coming and defect before me, and so on. Imagine that, however, player 1 does not defect in the first stage. What am I to infer? Was this a mistake or am I perhaps facing an irrational opponent? Backward induction requires that I never make such an inference, and hence I defect in stage 2.

Here is a better justification for defection in the centipede game, though. If player 1 doesn’t defect in the first stage, then I “try my best” to retain a belief in his rationality. That is, if it is possible for him to have some belief about my actions in the second stage which rationally justified his first stage action, then I must believe that he holds those beliefs. For example, he may believe that I believe he will continue again in the third stage, hence that I will continue in the second stage, hence he will continue in the first stage then plan to defect in the third stage. Given his beliefs about me, his actions in the first stage were rational. But if that plan to defect in stage three were his justification, then I should defect in stage two. He realizes I will make these inferences, hence he will defect in stage 1. That is, the backward induction outcome is justified by forward induction. Now, it can be proven that rationality and common “strong belief in rationality” as loosely explained above, along with a suitably rich type structure for all players, generates a backward induction outcome. But the epistemic justification is completely based on the equivalence between forward and backward induction under those assumptions, not on any epistemic justification for backward induction reasoning per se. I think that’s a fantastic result.

Final version, prepared for the new Handbook of Game Theory. I don’t see a version on RePEc IDEAS.

“The Axiomatic Structure of Empirical Content,” C. Chambers, F. Echenique & E. Shmaya (2013)

Here’s a particularly interesting article at the intersection of philosophy of science and economic theory. Economic theorists have, for much of the twentieth century, linked high theory to observable data using the technique of axiomatization. Many axiomatizations operate by proving that if an agent has such-and-such behavioral properties, their observed actions will encompass certain other properties, and vice versa. For example, demand functions over convex budget sets satisfy the strong axiom of revealed preference if and only if they are generated by the usual restrictions on preference.

You may wonder, however: to what extent is the axiomatization interesting when you care about falsification (not that you should care, necessarily, but if you did)? Note first that we only observe partial data about the world. I can observe that you choose apples when apples and oranges are available (A>=B or B>=A, perhaps strictly if I offer you a bit of money as well) but not whether you prefer apples or bananas when those are the only two options. This shows that a theory may be falsifiable in principle (I may observe that you prefer strictly A to B, B to C and C to A, violating transitivity, falsifying rational preferences) yet still make nonfalsifiable statements (rational preferences also require completeness, yet with only partial data, I can’t observe that you either weakly prefer apples to bananas, or weakly prefer bananas to apples).

Note something interesting here, if you know your Popper. The theory of rational preferences (complete and transitive, with strict preferences defined as the strict part of the >= relation) is universal in Popper’s sense: these axioms can be written using the “for all” quantifier only. So universality under partial observation cannot be all we mean if we wish to consider only the empirical content of a theory. And partial observability is yet harsher on Popper. Consider the classic falsifiable statement, “All swans are white.” If I can in principle only observe a subset of all of the swans in the world, then that statement is not, in fact, falsifiable, since any of the unobserved swans may actually be black.

What Chambers et al do is show that you can take any theory (a set of data generating processes which can be examined with your empirical data) and reduce it to stricter and stricter theories, in the sense that any data which would reject the original theory still reject the restricted theory. The strongest restriction has the following property: every axiom is UNCAF, meaning it can be written using only “for all” operators which negate a conjunction of atomic formulas. So “for all swans s, the swan is white” is not UNCAF (since it lacks a negation). In economics, the strict preference transitivity axiom “for all x,y,z, not x>y and y>z and z>x” is UNCAF and the completeness axiom “for all x,y, x>=y or y>=x” is not, since it is an “or” statement and cannot be reduced to the negation of a conjunction. It is straightforward to extend this to checking for empirical content relative to a technical axiom like continuity.

Proving this result requires some technical complexity, but the result itself is very easy to use for consumers and creators of axiomatizations. Very nice. The authors also note that Samuelson, in his rejoinder to Friedman’s awful ’53 methodology paper, more or less got things right. Friedman claimed that the truth of axioms is not terribly important. Samuelson pointed out that either all of a theory can falsified, in which case since the axioms themselves are always implied by a theory Friedman’s arguments are in trouble, or the theory makes some non-falsifiable claims, in which case attempts to test the theory as a whole are uninformative. Either way, if you care about predictive theories, you ought choose those the weakest theory that generates some given empirical content. In Chambers et al’s result, this means you better be choosing theories whose axioms are UNCAF with respect to technical assumptions. (And of course, if you are writing a theory for explanation, or lucidity, or simplicity, or whatever non-predictive goal you have in mind, continue not to worry about any of this!)

Dec 2012 Working Paper (no IDEAS version).

Paul Samuelson’s Contributions to Welfare Economics, K. Arrow (1983)

I happened to come across a copy of a book entitled “Paul Samuelson and Modern Economic Theory” when browsing the library stacks recently. Clear evidence of his incredible breadth are in the section titles: Arrow writes about his work on social welfare, Houthhaker on consumption theory, Patinkin on money, Tobin on fiscal policy, Merton on financial economics, and so on. Arrow’s chapter on welfare economics was particularly interesting. This book comes from the early 80s, which is roughly the end of social welfare as a major field of study in economics. I was never totally clear on the reason for this – is it simply that Arrow’s Possibility Theorem, Sen’s Liberal Paradox, and the Gibbard-Satterthwaite Theorem were so devastating to any hope of “general” social choice rules?

In any case, social welfare is today little studied, but Arrow mentions a number of interesting results which really ought be better known. Bergson-Samuelson, conceived when the two were in graduate school together, is rightfully famous. After a long interlude of confused utilitarianism, Pareto had us all convinced that we should dismiss cardinal utility and interpersonal utility comparisons. This seems to suggest that all we can say about social welfare is that we should select a Pareto-optimal state. Bergson and Samuelson were unhappy with this – we suggest individuals should have preferences which represent an order (complete and transitive) over states, and the old utilitarians had a rule which imposed a real number for society’s value of any state (hence an order). Being able to order states from a social point of view seems necessary if we are to make decisions. Some attempts to extend Pareto did not give us an order. (Why is an order important? Arrow does not discuss this, but consider earlier attempts at extending Pareto like Kaldor-Hicks efficiency: going from state s to state s’ is KH-efficient if there exist ex-post transfers under which the change is Paretian. Let person a value the bundle (1,1)>(2,0)>(1,0)>all else, and person b value the bundle (1,1)>(0,2)>(0,1)>all else. In state s, person a is allocated (2,0) and person b (0,1). In state s’, person a is allocated (1,0) and person b is allocated (0,2). Note that going from s to s’ is a Kaldor-Hicks improvement, but going from s’ to s is also a Kaldor-Hicks improvement!)

Bergson and Samuelson wanted to respect individual preferences – society can’t prefer s to s’ if s’ is a Pareto improvement on s in the individual preference relations. Take the relation RU. We will say that sRUs’ if all individuals weakly prefer s to s’. Not that though RU is not complete, it is transitive. Here’s the great, and non-obvious, trick. The Polish mathematician Szpilrajn has a great 1930 theorem which says that if R is a transitive relation, then there exists a complete relation R2 which extends R; that is, if sRs’ then sR2s’, plus we complete the relation by adding some more elements. This is not a terribly easy proof, it turns out. That is, there exists social welfare orders which are entirely ordinal and which respect Pareto dominance. Of course, there may be lots of them, and which you pick is a problem of philosophy more than economics, but they exist nonetheless. Note why Arrow’s theorem doesn’t apply: we are starting with given sets of preferences and constructing a social preference, rather than attempting to find a rule that maps any individual preferences into a social rule. There have been many papers arguing that this difference doesn’t matter, so all I can say is that Arrow himself, in this very essay, accepts that difference completely. (One more sidenote here: if you wish to start with individual utility functions, we can still do everything in an ordinal way. It is not obvious that every indifference map can be mapped to a utility function, and not even true without some type of continuity assumption, especially if we want the utility functions to themselves be continuous. A nice proof of how we can do so using a trick from probability theory is in Neuefeind’s 1972 paper, which was followed up in more generality by Mount and Reiter here at MEDS then by Chichilnisky in a series of papers. Now just sum up these mapped individual utilities, and I have a Paretian social utility function which was constructed entirely in an ordinal fashion.)

Now, this Bergson-Samuelson seems pretty unusable. What do we learn that we don’t know from a naive Pareto property? Here are two great insights. First, choose any social welfare function from the set we have constructed above. Let individuals have non-identical utility functions. In general, there is no social welfare function which is maximized by always keeping every individual’s income identical in all states of the world! The proof of this is very easy if we use Harsanyi’s extension of Bergson-Samuelson: if agents are Expected Utility maximizers, than any B-S social welfare function can be written as the weighted linear combination of individual utility functions. As relative prices or the social production possibilities frontier changes, the weights are constant, but the individual marginal utilities are (generically) not. Hence if it was socially optimal to endow everybody with equal income before the relative price change, it (generically) is not later, no matter which Pareto-respecting measure of social welfare your society chooses to use! That is, I think, an astounding result for naive egalitarianism.

Here’s a second one. Surely any good economist knows policies should be evaluated according to cost-benefit analysis. If, for instance, the summed willingness-to-pay for a public good exceeds the cost of the public good, then society should buy it. When, however, does a B-S social welfare function allow us to make such an inference? Generically, such an inference is only possible if the distribution of income is itself socially optimal, since willingness-to-pay depends on the individual budget constraints. Indeed, even if demand estimation or survey evidence suggests that there is very little willingness-to-pay for a public good, society may wish to purchase the good. This is true even if the underlying basis for choosing the particular social welfare function we use has nothing at all to do with equity, and further since the B-S social welfare function respects individual preferences via the Paretian criterion, the reason we build the public good also has nothing to do with paternalism. Results of this type are just absolutely fundamental to policy analysis, and are not at all made irrelevant by the impossibility results which followed Arrow’s theorem.

This is a book chapter, so I’m afraid I don’t have an online version. The book is here. Arrow is amazingly still publishing at the age of 91; he had an interesting article with the underrated Partha Dasgupta in the EJ a couple years back. People claim that relative consumption a la Veblen matters in surveys. Yet it is hard to find such effects in the data. Why is this? Assume I wish to keep up with the Joneses when I move to a richer place. If I increase consumption today, I am decreasing savings, which decreases consumption even more tomorrow. How my desire to change consumption today if I have richer peers then depends on that dynamic tradeoff, which Arrow and Dasgupta completely characterize.

“The Meaning of Utility Measurement,” A. Alchian (1953)

Armen Alchian, one of the dons from UCLA’s glory days, passed away today at 98. His is, for me, a difficult legacy to interpret. On the one hand, Alchian-Demsetz 1972 is among the most famous economics papers ever written, and it can fairly be considered the precursor to mechanism design, the most important new idea in economics in the past 50 years. People produce more by working together. It is difficult to know who shirks when we work as a team. A firm gives a residual claimant (an owner) who then has an incentive to monitor shirking, and as only one person needs to monitor the shirking, this is much less costly than a market where each member of the team production would need somehow to monitor whether other parts of the team shirk. Firms are deluded if they think that they can order their labor inputs to do whatever they want – agency problems exist both within and outside the firm. Such an agency theory of the firm is very modern indeed. That said, surely this can’t explain things like horizontally integrated firms, with different divisions producing wholly different products (or, really, any firm behavior where output is a separable function of each input in the firm).

Alchian’s other super famous work is his 1950 paper on evolution and the firm. As Friedman would later argue, Alchian suggested that we are justified treating firms as if they are profit maximizers when we do our analyses since the nature of competition means that non-profit maximizing firms will disappear in the long run. I am a Nelson/Winter fan, so of course I like the second half of the argument, but if I want to suggest that firms partially seek opportunities and partially are driven out by selection (one bit Lamarck, one bit Darwin), then why not just drop the profit maximization axiom altogether and try to write a parsimonious description of firm behavior which doesn’t rely on such maximization?

It turns out that if you do the math, profit maximization is not generally equivalent to selection. Using an example from Sandroni 2000, take two firms. There are two equally likely states of nature, Good and Bad. There are two things a firm can do, the risky one, which returns profit 3 in good states and 0 in bad states, and a risk-free one, which always returns 1. Maximizing expected profit means always investing all capital in the risky state, hence eventually going bankrupt. A firm who doesn’t profit maximize (say, it has incorrect beliefs and thinks we are always in the Bad state, hence always takes the risk-free action) can survive. This example is far too simple to be of much worth, but it does at least remind us of lesson in the St. Petersburg paradox: expected value maximization and survival have very little to do with each other.

More interesting is the case with random profits, as in Radner and Dutta 2003. Firms invest their capital stock, choosing some mean-variance profits pair as a function of capital stock. The owner can, instead of reinvesting profits into the capital stock, pay out to herself or investors. If the marginal utility of a dollar of capital stock falls below a dollar, the profit-maximizing owner will not reinvest that money. But a run of (random) losses can drive the firm to bankruptcy, and does so eventually with certainty. A non-profit maximizing firm may just take the lowest variance earnings in every period, pay out to investors a fraction of the capital stock exactly equal to the minimum earnings that period, and hence live forever. But why would investors ever invest in such a firm? If investment demand is bounded, for example, and there are many non profit-maximizing firms from the start, it is not the highest rate of return but the marginal rate of return which determines the market interest rate paid to investors. A non profit-maximizer that can pay out to investors at least that much will survive, and all the profit maximizers will eventually fail.

The paper in the title of this post is much simpler: it is merely a very readable description of von Neumann expected utility, when utility can be associated with a number and when it cannot, and the possibility of interpersonal utility comparison. Alchian, it is said, was a very good teacher, and from this article, I believe it. What’s great is the timing: 1953. That’s one year before Savage’s theory, the most beautiful in all of economics. Given that Alchian was associated with RAND, where Savage was fairly often, I imagine he must have known at least some of the rudiments of Savage’s subjective theory, though nothing appears in this particular article. 1953 is also two years before Herbert Simon’s behavioral theory. When describing the vN-M axioms, Alchian gives situations which might contradict each, except for the first, a complete and transitive order over bundles of goods, an assumption which is consistent with all but “totally unreasonable behavior”!

1953 AER final version (No IDEAS version).

“Until the Bitter End: On Prospect Theory in a Dynamic Context,” S. Ebert & P. Strack (2012)

Let’s kick off job market season with an interesting paper by Sebastian Ebert, a post-doc at Bonn, and Philipp Strack, who is on the job market from Bonn (though this doesn’t appear to be his main job market paper). The paper concerns the implications of Tversky and Kahneman’s prospect theory is its 1992 form. This form of utility is nothing obscure: the 1992 paper has over 5,000 citations, and the original prospect theory paper has substantially more. Roughly, cumulative prospect theory (CPT) says that agents have utility which is concave above a reference point, convex below it, with big losses and gains that occur with small probability weighed particularly heavily. Such loss aversion is thought to explain, for example, the simultaneous existence of insurance and gambling, or the difference in willingness to pay for objects you possess versus objects you don’t possess.

As Machina, among others, pointed out a couple decades ago, once you leave expected utility, you are definitely going to be writing down preferences that generate strange behavior at least somewhere. This is a direct result of Savage’s theorem. If you are not an EU-maximizer, then you are violating at least one of Savage’s axioms, and those axioms in their totality are proven to avoid many types of behavior that we find normatively unappealing such as falling for the sunk cost fallacy. Ebert and Strack write down a really general version of CPT, even more general than the rough definition I gave above. They then note that loss aversion means I can always construct a right-skewed gamble with negative expected payout that the loss averse agent will accept. Why? Agents like big gains that occur with small probability. Right-skew the gamble so that a big gain occurs with a tiny amount of probability, and otherwise the agent loses a tiny amount. An agent with CPT preferences will accept this gamble. Such a gamble exists at any wealth level, no matter what the reference point. Likewise, there is a left-skewed, positive expected payoff gamble that is rejected at any wealth level.

If you take a theory-free definition of risk aversion to mean “Risk-averse agents never accept gambles with zero expected payoff” and “Risk-loving agents always accept a risk with zero expected payoff”, then the theorem in the previous paragraph means that CPT agents are neither risk-averse, nor risk-loving, at any wealth level. This is interesting because a naive description of the loss averse utility function is that CPT agents are “risk-averse above the reference point, and risk-loving below it”. But the fact that small probability events are given more weight, in Ebert and Strack’s words, dominates whatever curvature the utility function possesses when it comes to some types of gambles.

So what does this mean, then? Let’s take CPT agents into a dynamic framework, and let them be naive about their time inconsistency (since they are non EU-maximizers, they will be time inconsistent). Bring them to a casino where a random variable moves with negative drift. Give them an endowment of money and any reference point. The CPT agent gambles at any time t as long as she has some strategy which (naively) increases her CPT utility. By the skewness result above, we know she can, at the very least, gamble a very small amount, plan to stop if I lose, and plan to keep gambling if I win. There is always such a bet. If I do lose, then tomorrow I will bet again, since there is a gamble with positive expected utility gain no matter my wealth level. Since the process has negative drift, I will continue gambling until I go bankrupt. This result isn’t relying on any strange properties of continuous time or infinite state spaces; the authors construct an example on a 37-number roulette wheel using the original parameterization of Kahneman and Tversky which has the CPT agent bet all the way to bankruptcy.

What do we learn? Two things. First, a lot of what is supposedly explained by prospect theory may, in fact, be explained by the skewness preference which the heavy weighting on low probability events in CPT, a fact mentioned by a number of papers the authors cite. Second, not to go all Burke on you, but when dealing with qualitative models, we have good reason to stick to the orthodoxy in many cases. The logical consequences of orthodox models will generally have been explored in great depth. The logical consequences of alternatives will not have been explored in the same way. All of our models of dynamic utility are problematic: expected utility falls in the Rabin critique, ambiguity aversion implies sunk cost fallacies, and prospect theory is vulnerable in the ways described here. But any theory which has been used for a long time will have its flaws shown more visibly than newer, alternative theories. We shouldn’t mistake the lack of visible flaws for their lack more generally.

SSRN Feb. 2012 working paper (no IDEAS version).

“Das Unsicherheitsmoment in der Wirtlehre,” K. Menger (1934)

Every economist surely knows the St. Petersburg Paradox described by Daniel Bernoulli in 1738 in a paper which can fairly claim to be the first piece of theoretical economics. Consider a casino offering a game of sequential coinflips that pays 2^(n-1) as a payoff if the first heads arrives on the nth flip of the coin. That is, if there is a heads on the first flip, you receive 1. If there is a tails on the first flip, and a heads on the second, you receive 2, and 4 if TTH, and 8 if TTTH, and so on. It is quite immediate that this game has expected payoff of infinity. Yet, Bernoulli points out, no one would pay anywhere near infinity for such a game. Why not? Perhaps they have what we would now call logarithmic utility, in which case I value the gamble at .5*ln(1)+.25*ln(2)+.125*ln(4)+…, a finite sum.

Now, here’s the interesting bit. Karl Menger proved in the 1927 that the standard response to the St. Petersburg paradox is insufficient (note that Karl with a K is the mathematically inclined son and mentor to Morganstern, rather than the relatively qualitative father, Carl, who somewhat undeservingly joined Walras and Jevons on the Mt. Rushmore of Marginal Utility). For instance, if the casino pays out e^(2^n-1) rather than 2^(n-1), then even an agent with logarithmic utility have infinite expected utility from such a gamble. This, nearly 200 years after Bernoulli’s original paper! Indeed, such a construction is possible for any unbounded utility function; let the casino pay out U^-1(2^(n-1)) when the first heads arrives on the nth flip, where U^-1 is inverse utility.

Things are worse, Menger points out. One can construct a thought experiment where, for any finite amount C and an arbitrarily small probability p, there is a bounded utility function where an agent will prefer the gamble to win some finite amount D with probability p to getting a sure thing of C [Sentence edited as suggested in the comments.] So bounding the utility function does not kill off all paradoxes of this type.

The 1927 lecture and its response are discussed in length in Rob Leonard’s “Von Neumann, Morganstern, and the Creation of Game Theory.” Apparently, Oskar Morganstern was at the Vienna Kreis where Menger first presented this result, and was quite taken with it, a fact surely interesting given Morganstern’s later development of expected utility theory. Indeed, one of Machina’s stated aims in his famous paper on EU with the Independence Axiom is providing a way around Menger’s result while salvaging EU analysis. If you are unfamiliar with Machina’s paper, one of the most cited in decision theory in the past 30 years, it may be worthwhile to read the New School HET description of the “fanning out” hypothesis which relates Machina to vN-M expected utility. (Unfortunately, the paper above is both gated, and in German, as the original publication was in the formerly-famous journal Zeitschrift fur Nationalokonomie. The first English translation is in Shubik’s festschrift for Morganstern published in 1967, but I don’t see any online availability.)

“A Bayesian Model of Risk and Uncertainty,” N. al-Najjar & J. Weinstein (2012)

In a Bayesian world with expected utility maximizers, you have a prior belief on the chance that certain events will occur, and you maximize utility subject to those beliefs. But what if you are “uncertain” about what your prior even is? Perhaps you think with 60 percent probability, peace negotiations will commence and there will be a .5 chance of war and a .5 chance of peace, but with 40 percent probability, war is guaranteed to occur. It turns out these types of compound lotteries don’t affect your decision if you’re just making a single choice: simply combine the compound lottery and use that as your prior. In this case, you think war will occur with .6*.5+.4*1=.7 probability. That is, the Bayesian world is great for discussing risk – decisionmaking with concave utility and known distributions – not that useful for talking about one-shot Knightian uncertainty, or decisionmaking when the distributions are not well-known.

al-Najjar and Weinstein show, however, that this logic does not hold when you take multiple decisions that depend on a parameter that is common (or at least correlated) across those decisions. Imagine that a stock has a daily return determined by some IID process which is bought by a risk-averse agent, and imagine that the agent doesn’t have a single prior about that parameter, but rather a prior over the set of possible priors. For instance, as above, with probability .6 you have a .5 chance of a 1 percent increase and a .5 chance of a 1 percent decrease, but with probability .4, a 1 percent increase is assured. Every period, I can update my “prior over priors”. Does the logic about the compound lottery collapsing still hold, or does this uncertainty matter for decisionmaking?

If utility is linear or separable over time, then uncertainty doesn’t matter, but otherwise it does. Why? Call the prior over your priors “uncertainty.” Mathematically, the expected utility is a double integral: the outer integral is over possible priors with respect to your uncertainty, and the inner integral is just standard expected utility over N time periods with respect to each prior currently being summed in the outer integral. In the linear or separable utility case, I can swap the position of the integrals with the summation of utility over time, making the problem equivalent to adding up N one-period decision problems; as before, having priors over your prior when only one decision is being made cannot affect the decision you make, since you can just collapse the compound lottery.

If utility is not linear or separable over time, uncertainty will affect your decision. In particular, with concave utility, you will be uncertainty averse in addition to being risk averse. Al-Najjar and Weinstein use a modified Dirichlet distribution to talk about this more concretely. In particular, assuming a uniform prior-over-priors is actually equivalent to assuming very little uncertainty: the uniform prior-over-priors will respond very slowly to information learned during the first few periods. Alternatively, if you have a lot of uncertainty (a low Dirichlet parameter), your prior-over-priors, and hence your decisions, will change rapidly in the first few periods.

So what’s the use of this model? First, it allows you to talk about dynamic uncertainty without invoking any of the standard explanations for ambiguity – the problems with the ambiguity models are discussed in a well-known 2009 article by the authors of the present paper. If you’re, say, an observer of people’s behavior on the stock market, and see actions in some sectors that suggests purchase variability that far exceeds the known ex-post underlying variability of the asset, you might want to infer that the prior-over-priors exhibited a lot of uncertainty during the time examined; the buyers were not necessarily irrational. In particular, during regime shifts or periods with new financial product introduction, even if the ex-post level of risk does not change, assets may move with much more variance than expected due to the underlying uncertainty. Alternatively, if new assets whose underlying parameters are likely to be subject to much Knightian uncertainty, this model gives you a perfectly Bayesian explanation for why returns on that asset are higher than seems justified given known levels of risk aversion.

December 2011 Working Paper

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